Quantitative Biology > Quantitative Methods

Title:
Asymptotic scaling properties of the posterior mean and variance in the Gaussian scale mixture model

Abstract: The Gaussian scale mixture model (GSM) is a simple yet powerful probabilistic
generative model of natural image patches. In line with the well-established
idea that sensory processing is adapted to the statistics of the natural
environment, the GSM has also been considered a model of the early visual
system, as a reasonable "first-order" approximation to the internal model that
the primary visual cortex (V1) inverts. According to this view, neural
activities in V1 represent the posterior distribution under the GSM given a
particular visual stimulus. Indeed, (approximate) inference under the GSM has
successfully accounted for various nonlinearities in the mean (trial-average)
responses of V1 neurons, as well as the dependence of (across-trial) response
variability with stimulus contrast found in V1 recordings. However, previous
work almost exclusively relied on numerical simulations to obtain these
results. Thus, for a deeper insight into the realm of possible behaviours the
GSM can (and cannot) exhibit and predict, here we present analytical
derivations for the limiting behaviour of the mean and (co)variance of the GSM
posterior at very low and high contrast levels. These results should guide
future work exploring neural circuit dynamics appropriate for implementing
inference under the GSM.